Week 4: Clustering and classification

Data exploration and linear discriminant analysis

This week, the data set under scrutiny contains housing values in the suburbs of Boston in the late 1970s (more information about the data set can be found here). It is part of the MASS library in R. Let’s have a look at its variables:

## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...

The data set consists of 506 neighbourhoods in Boston and for each there are 14 explanatory variables, such as crime rate per capita and nitric oxides concentration. Let’s then have a closer look at the variables and visualize them.

##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08205   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

All of the variables are numerical. The chas variable is the only binary one, while the others have ranges of varying magnitudes. Some of the variable pairs are very strongly correlated, such as (rad, tax) and (nox, dis), while many pairs are relatively independent of each other.

In order to be able to do proper linear discriminant analysis (LDA) later, all the variables need to be scaled. This is done for each variable by subtracting the mean and dividing by the standard deviation. Here is how the variables look after scaling:

##       crim                 zn               indus              chas        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109   Median :-0.2723  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648  
##       nox                rm               age               dis         
##  Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658  
##  1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049  
##  Median :-0.1441   Median :-0.1084   Median : 0.3171   Median :-0.2790  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617  
##  Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566  
##       rad               tax             ptratio            black        
##  Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033  
##  1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049  
##  Median :-0.5225   Median :-0.4642   Median : 0.2746   Median : 0.3808  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332  
##  Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406  
##      lstat              medv        
##  Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 3.5453   Max.   : 2.9865

Now all the variables are of comparable magnitude. Next let’s take the crim variable, which is the per capita crime rate, and turn it into a categorical variable crime using the quantiles as break points. The four categories of approximately equal size are named low, med_low, med_high and high:

## 
##      low  med_low med_high     high 
##      127      126      126      127

Before we attempt to classify the data with LDA, the data is divided into train and test sets with 80/20 split:

## Train set size: 404
## Test set size: 102

With the scaled training set, crime is used as a target variable for the LDA and all the other variables are predictor variables.

Let’s see how well our LDA model fares when evaluated on the test set:

test_y <- test$crime
test_x <- dplyr::select(test, -crime)

lda.pred <- predict(lda.fit, newdata = test_x)
table(correct = test_y, predicted = lda.pred$class)

##           predicted
## correct    low med_low med_high high
##   low       23      10        1    0
##   med_low    3      14        4    0
##   med_high   0      11       12    1
##   high       0       0        1   22

The table shows that the neighbourhoods with the highest crime rates are identified very reliably, while the neighbourhoods with lower crime rates cannot be classified as successfully. Most notably, the LDA cannot distinguish medium low and medium high crime rate areas convincingly based on these explanatory features, and one neighbourhood with low crime rate was even classified as a medium high crime rate area.

k-means clustering

Let’s now for a moment forget about the LDA that was done and try to also cluster the original data by using the k-means clustering algorithm. The data is first scaled similary as earlier. Let’s have a look at the summary of the Euclidian distances in the scaled data set:

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

The minimum and maximum values indicate that some data points are indeed very close to each other when compared to the mean distance, but there’s also considerable distances between other points. This signifies that the data set is not homogeneous and the data points form some sorts of clusters. The optimal number of clusters can be investigated by running the k-means algorithm for a range of k and calculating the Within-Cluster-Sum-of-Squares (WCSS) value for each k. The plot for WCSS as a function k is shown below.

By design, the WCSS value decreases monotonously as k is increased. The optimal value for k can be found by considering the so-called elbow point, where there is a biggest drop in WCSS between two values. This implies that perhaps k = 2 is the optimal value, but k = 3 should work fine as well. After that, the slope becomes less steep at each point. The clusters can be visualized by looking at the variables and coloring the different groups. Here k = 3 was used:

The three categories certainly form their own clusters, though it is quite difficult to interpret how well it works because these plots are 2-dimensional, while the clusters exist in 14-dimensional space. Let us then combine LDA and the k-means clustering to see if LDA is capable of finding the clusters. All the 14 explanatory variables are used as predictors, while the thee cluster categories are used as a target.

From this plot we can see that as was observed earlier, there is clearly one cluster of data points separate from the majority of the data points. The rad and tax variables are the most influential separators for the clusters. It was seen in the earlier LDA biplot as well that rad was important in separating the high crime rate neighbourhoods from the others. We can verify whether cluster 1 and the high crime rate areas are indeed the same clusters by comparing two 3D plots of the LDA, one labeled with the crime categories and another with the k-means clusters. (You can rotate the plots by dragging them!)

It appears that that cluster 1 practically corresponds to the high crime rate neighbourhoods. The k-means clustering algorithm also did reasonable job with identifying the low crime rate neighbourhoods (cluster 2) and the medium low crime rate neighbourhoods (cluster 3). Of course the k-means algorithm operated with one category less than the LDA, so they cannot have a one-to-one correspondence.